The net force acting on the gate element of width dy at a depth \(y\) from the surface of the fluid, is

\(dy  =\left( p _0+\rho_{ g } y - p _0\right) \times 1 dy\)

\(=\rho \text { gydy }\)

Torque about the hinge is

\(d \tau=\operatorname{pgydy} \times\left(\frac{1}{2}- y \right)\)

Net torque experienced by the gate is

\(\tau_{\text {net }}=\int d \tau+ F \times \frac{1}{2}=0\)

\(\Rightarrow F =\frac{\rho g }{6}\)